Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Meyer-Itô Formula for Stable Processes via Fractional Calculus

Alejandro Santoyo Cano, Gerónimo Uribe Bravo

Abstract

The infinitesimal generator of a one-dimensional strictly $α$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $α$ and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $α\in (0,1)$ and the recurrent potential if $α\in (1,2)$. The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $α$. One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for $|X_t - x|^γ$ with $X$ a recurrent strictly stable process of index $α$ and $γ\in (α-1,α)$, generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case.

A Meyer-Itô Formula for Stable Processes via Fractional Calculus

Abstract

The infinitesimal generator of a one-dimensional strictly -stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if and the recurrent potential if . The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order . One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for with a recurrent strictly stable process of index and , generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case.
Paper Structure (4 sections, 20 theorems, 105 equations)

This paper contains 4 sections, 20 theorems, 105 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Preliminaries: stable processes and fractional operators
  3. Proof of the main results
  4. Trigonometric results

Key Result

Proposition 1.4

Let $\alpha \in (0,2)\setminus \left\{1\right\}$, $c_-, c_+ \geq 0$, not both zero. If $X\sim S_{\alpha}\left( c_-, c_+ \right)$, then the domain of the infinitesimal generator $\mathcal{L}$ of $X$ contains $\mathcal{S}(\mathbb{R})$. For $\varphi \in \mathcal{S}(\mathbb{R})$, we have: where $M_\pm = c_\pm \Gamma(-\alpha)$.

Theorems & Definitions (44)

  • Definition 1.1: Strictly stable process
  • Definition 1.2: Riemann-Liouville fractional operators
  • Remark 1.3
  • Proposition 1.4: Infinitesimal generator
  • Remark 1.5
  • Definition 1.6: Lizorkin space
  • Proposition 1.7
  • Theorem 1: Inverse of the Infinitesimal Generator
  • Definition 1.8: Occupational local time
  • Definition 1.9
  • ...and 34 more